Hyder, Ali and Martinazzi, Luca.
(2014)
* Conformal Metrics on $R^{2m}$ with Constant Q-Curvature, Prescribed Volume and Asymptotic Behavior.*

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Official URL: https://edoc.unibas.ch/70018/

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## Abstract

We study the solutions $u\in C^\infty(R^{2m})$ of the problem $(-\Delta)^m u= Qe^{2mu}$, where $Q=\pm (2m-1)!$, and $V :=\int_{R^{2m}}e^{2mu}dx <\infty$, particularly when $m>1$. This corresponds to finding conformal metrics $g_u:=e^{2u}|dx|^2$ on $R^{2m}$ with constant Q-curvature $Q$ and finite volume $V$. Extending previous works of Chang-Chen, and Wei-Ye, we show that both the value $V$ and the asymptotic behavior of $u(x)$ as $|x|\to \infty$ can be simultaneously prescribed, under certain restrictions. When $Q=(2m-1)!$ we need to assume $V<vol(S^{2m})$, but surprisingly for $Q=-(2m-1)!$ the volume $V$ can be chosen arbitrarily.

Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Analysis (Martinazzi) 12 Special Collections > Preprints Fachbereich Mathematik |
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UniBasel Contributors: | Hyder, Ali and Martinazzi, Luca |

Item Type: | Preprint |

Publisher: | Universität Basel |

Last Modified: | 12 May 2019 23:09 |

Deposited On: | 28 Mar 2019 09:52 |

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